# Questions tagged [ideals]

An ideal is a subset of ring such that it is possible to make a quotient ring with respect to this subset. This is the most frequent use of the name ideal, but it is used in other areas of mathematics too: ideals in set theory and order theory (which are closely related), ideals in semigroups, ideals in Lie algebras.

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### Show that $\frac{\mathbb{H}[\mathbb{Z}]}{ I_p}$ is isomorph in ${\mathbb{H}[\mathbb{Z}_p]}$

Consider ${I_p}$ = ${a_0 + a_1 i + a_2 j + a_3 k \in \mathbb{H}[\mathbb{Z}]}$, so that $p | a_i$
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### Does every evaluation map $g: \Bbb{Z}[X,Y]/(\ker \pi) \to \Bbb{Z}$ factor some evaluation map $f:\Bbb{Z}[X,Y] \to \Bbb{Z}$?

Let $\Bbb{Z}[X,Y]$ be the polynomial ring. I know that every evaluation ring hom $f: \Bbb{Z}[X,Y] \to \Bbb{Z}$ is determined by where you send $X$ and $Y$. Let $I = (X^2 - Y^3)$ for example, but it ...
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### Is the ideal $\langle x^2 + 1\rangle$ maximal in $\mathbb{Z}_3[x]$?

Is the ideal $\langle x^2 + 1\rangle$ maximal in $\mathbb{Z}_3[x]$? I am going about this by trying to prove that $\frac{\mathbb{Z}_3[x]}{\langle x^2 + 1 \rangle}$ is a field. I can prove commutative ...
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### Showing $\langle x,y \rangle$ is a prime ideal of $K[x,y]$

The biggest thing that it stopping me from progressing in this question is the fact that I have two elements in the ideal, this topic it still very new to me. I can show that $I=\langle x \rangle$ and ...
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